Pivots and Echelon Form

The Essentials

A pivot is the first non-zero entry in each row in a matrix. Row echelon form is when every number below a pivot is zero, and the pivots move down and to the right, with the all zero rows on the bottom. The following is an example of a matrix in echelon form:

\[ \begin{bmatrix} 2 & * & * & * & *\\ 0 & 0 & 3 & * & *\\ 0 & 0 & 0 & 1 & *\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

A matrix in echelon form is in reduced echelon form if each of the pivots has a value of 1 and the components above the pivots are 0 in addition to the ones below the pivots. This is an example of a matrix in reduced row echelon form:

\[ \begin{bmatrix} 1 & * & 0 & 0 & *\\ 0 & 0 & 1 & 0 & *\\ 0 & 0 & 0 & 1 & *\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

Note that each matrix can reduce differently to echelon form, but every matrix has a unique reduced echelon form. For a linearly independent square matrix, the reduced echelon form is the identity matrix of the appropriate size.

Example

Reduce this matrix to reduced echelon form:

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 2 & 0 & 1 & -2 & 5\\ 0 & -1 & 3 & 1 & 1 \end{bmatrix} \]

\( R_2=R_2-2R_1 \)

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 0 & -2 & 1 & 6 & 3\\ 0 & -1 & 3 & 1 & 1 \end{bmatrix} \]

\( R_2 \longleftrightarrow R_3 \)

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 0 & -1 & 3 & 1 & 1\\ 0 & -2 & 1 & 6 & 3 \end{bmatrix} \]

\( R_3=R_3-2R_2 \)

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 0 & -1 & 3 & 1 & 1\\ 0 & 0 & -5 & 4 & 1 \end{bmatrix} \]

At this point the matrix is in row echelon form.
\( R_2=5R_2 \)
\( R_3=3R_3 \)

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 0 & -5 & 15 & 5 & 5\\ 0 & 0 & -15 & 12 & 3 \end{bmatrix} \]

\( R_2=R_2+R_3 \)

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 0 & -5 & 0 & 17 & 8\\ 0 & 0 & -15 & 12 & 3 \end{bmatrix} \]

\( R_2=-R_2/5 \)
\( R_3=-R_3/15 \)

\[ \begin{bmatrix} 1 & 1 & 0 & -4 & 1\\ 0 & 1 & 0 & -17/5 & -8/5\\ 0 & 0 & 1 & -4/5 & -1/5 \end{bmatrix} \]

\( R_1=R_1-R_2 \)

\[ \begin{bmatrix} 1 & 0 & 0 & -3/5 & 13/5\\ 0 & 1 & 0 & -17/5 & -8/5\\ 0 & 0 & 1 & -4/5 & -1/5 \end{bmatrix} \]

Which is in reduced row echelon form.

Practice

Reduce this matrix to reduced row echelon form:

\[ \begin{bmatrix} 2 & 3 & 5\\ -1 & 3 & 9\\ -2 & -4 & 11 \end{bmatrix} \]

Solution:

\[ \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \]